Half of a One-sided Paper = ?

What do you get when you cut a Mobius strip in half lengthwise? The answer may surprise you!

This is a simple exercise but may be both motivating and confounding.

For children who have not played with Mobius strips before, start by introducing the idea of the strip - here are a few ways:

1) ask the child, 'is there a way to make this strip of paper have only one side?'

or

2) make one without explanation by simply cutting a strip of paper, twisting it, and taping the ends. Ask your child to draw a line along the shape as far as they can.

Once they have played with and explored a regular Mobius strip, ask what would happen if you were to cut it lengthwise? Would you get two strips? Demonstrate and experiment with the properties of the result.

We got the idea for this exercise from Critical Thinking Puzzles by Michael DiSpezio.

Johann Benedict Listing, who discovered the strip independently at the same time as Mobius, also experimented with higher order twists called 'paradromic rings'. These are defined as 'Rings produced by cutting a strip that has been given m half twists and been re-attached into n equal strips (Ball and Coxeter 1987, pp. 127-128). ' - http://mathworld.wolfram.com/ParadromicRings.html. Now there is a project for the curious...

Tip for understanding the cut mobius strip - hover over the area below:
Can you make something similar to the cut mobius strip by taping and twisting a long strip of paper without cutting it further? How many twists?

To understand any particular twisted paper shape, cut across the strip but hold the cut ends together. Then untwist slowly, counting the twists (or half-turns) until it becomes a normal ring. If you can unmake and re-make a shape, you have a grasp of its topology.

This exercise is a great opportunity to model curiousity in the face of the unknown! Show your children how to have fun investigating carefully something that initially seems confusing.